# Research

“Quantum theory provides us with a striking illustration of the fact that we can fully understand a connection though we can only speak of it in images and parables.”

Werner Heisenberg

## General Research Interests

As a condensed matter theorist, I'm fascinated by the fundamental puzzles that the quantum world still has to offer, while quantum phenomena already directly impact our daily life. In my research, I explore topological states of matter and the potential of non-equilibrium systems. Currently, I am investigating how to use cooling for quantum computation.

## In plain English

## Condensed Matter Theory

Have you heard about Schrödinger's cat? It is a beautiful illustration of what quantum means - the possibility to be in more than one state at the same time. For example, an electron could simultaneously be at two different locations. If you zoom into our world close enough, you discover quantum mechanical objects, like atoms, electrons, and photons (the particles of light). In condensed matter theory, we study how the interaction of atoms, electrons, and light leads to macroscopic phenomena like magnetism and superconductivity. Another very exciting question is how you can design a quantum computer with these principles at hand.

## Topology

We like to talk in pictures when we explain topology. Topology is a field of mathematics. It describes the properties of objects that don't change under smooth deformation. Look at the heart-shaped doughnut above and imagine it is made out of modeling clay. You may deform it into a round doughnut. But if you want to deform the doughnut into a ball, you have to close the hole. Therefore the doughnut and the ball are very different in a topological sense. They have a different number of holes. In physical systems, we don't talk about holes anymore. But we can also identify properties that stay constant if we smoothly change the system parameters.

## Out of equilibrium physics

If you enter an airplane, you can explore new worlds, get new perspectives, and discover unexplored ground. Similarly, you find phenomena that you cannot find in equilibrium systems if you change system parameters in time. Since such a perturbation brings the system out of its favorite resting place, we say it is now in a state out of equilibrium.

## Publications

Quantum computing and non-equilibrium dynamics are two very rapidly developing and entangled fields. With the constant application of gates and measurements, quantum computers are always out of equilibrium. Furthermore, the current noisy quantum chips are prone to errors and dissipation, and are effectively described as an open system. At the same time, the simulation of the long-time dynamics of quantum systems is one of the most promising applications of a quantum computer. This thesis spans a range of topics, from qubit design to quantum algorithms. The key feature of our proposed qubit is imprinted by non-equilibrium dynamics. Namely, we address how to enhance the basic building block of a Majorana-based quantum computer by periodic driving. The so-called Floquet Majorana box qubits can host not only Majorana zero modes with quasi-energy zero but also Floquet Majoranas with an energy corresponding to half of is the driving frequency. This allows us to encode three topological logical qubits in one box. However, a standard adiabatic state preparation protocol fails, and we argue that this instability is a generic and fundamental feature of a Floquet superconductor. Instead, we show that it can be successfully operated using a frequency-sweep protocol, even in the presence of interactions. On the topic of quantum algorithms, we propose a scalable and robust protocol that prepares low-energy states of arbitrary gapped Hamiltonians, without prior knowledge about the target state. By using a fraction of the qubits to mimic a low-entropy bath, the protocol effectively cools the system to its low-energy state. The cyclic operation of the protocol is a key advantage since it leads to a robust "coolability" in the presence of noise. We investigate the performance of the protocol for systems with trivial and topological excitations. Because topological excitations are notoriously difficult to remove, the coolability can help to detect topological order. Finally, we discuss how this cooling protocol can be implemented on a gate-based quantum computer. While we experimentally tested the protocol on only very few qubits, the promising results suggest that the cooling protocol will be valuable for the preparation of more complex many-body states on future quantum computers.

## Programmable adiabatic demagnetization for systems with trivial and topological excitations

preprint arXiv:2210.17256

Anne Matthies, Mark Rudner, Achim Rosch and Erez Berg

We propose a simple, robust protocol to prepare a low-energy state of an arbitrary Hamiltonian on a quantum computer or programmable quantum simulator. The protocol is inspired by the adiabatic demagnetization technique, used to cool solid-state systems to extremely low temperatures. A fraction of the qubits (or spins) is used to model a spin bath that is coupled to the system. By an adiabatic ramp down of a simulated Zeeman field acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state is achieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol's performance in the presence of noise and show how the information from the measurement of the bath spins can be used to monitor the cooling process. The performance of the algorithm depends on the nature of the excitations of the system; systems with non-local (topological) excitations are more difficult to cool than those with local excitations. We explore the possible mitigation of this problem by trapping topological excitations. arXiv:2210.17256

## Stability of Floquet Majorana Box Qubits

published in Physical Review Letters

Anne Matthies, Jinhong Park, Erez Berg and Achim Rosch

In one-dimensional topological superconductors driven periodically with the frequency ω, two types of topological edge modes may appear, the well-known Majorana zero mode and a Floquet Majorana mode located at the energy ω/2. We investigate two Josephson-coupled topological quantum wires in the presence of Coulomb interactions, forming a so-called Majorana box qubit. An oscillating gate voltage can induce Floquet Majorana modes in both wires. This allows encoding 3 qubits in a sector with fixed electron parity. If such a system is prepared by increasing the amplitude of oscillations adiabatically, it is inherently unstable as interactions resonantly create quasi particles. This can be avoided by using instead a protocol where the oscillation frequency is increased slowly. In this case, one can find a parameter regime where the system remains stable. arXiv:2110.05281

## Control of competing superconductivity and charge order by nonequilibrium currents

published in PRB Rapid Communication

Anne Matthies, Jiajun Li and Martin Eckstein

We study the competing charge-density-wave and superconducting order in the attractive Hubbard model under a voltage bias, using steady-state nonequilibrium dynamical mean-field theory. We show that the charge-density wave is suppressed in a current-carrying nonequilibrium steady state. This effect is beyond a simple Joule-heating mechanism and a “supercooled” metallic state is stabilized at a nonequilibrium temperature lower than the equilibrium superconducting Tc . On the other hand, a current-carrying superconducting state is dissipationless and thus not subject to the same nonthermal suppression, and can therefore nucleate out of the supercooled metal, e.g., in a resistive switching experiment. The fact that an electric current can change the relative stability of different phases compared to thermal equilibrium, even when a system appears locally thermal due to electron-electron scattering, provides a general perspective to control intertwined orders out of equlibrium. arXiv:1804.09608